#4380244

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#535	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(5,6,7,8)(9,10,13,15)(11,16,14,12)(19,23)(21,24);(9,11)(10,12)(13,14)(15,16)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(11,14)(12,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,14)(13,15);(9,12,13,16)(10,14,15,11)(17,19)(18,21)(20,23)(22,24)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4287440);